Related papers: Arithmetic properties of $(\ell,m)$-regular colore…

Let $b_{\ell, k}(n), b_{\ell, k, r}(n)$ count the number of $(\ell, k)$, $(\ell, k, r)$-regular partitions respectively. In this paper we shall derive infinite families of congruences for $b_{\ell, k}(n)$ modulo $2$ when $ (\ell, k) =…

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0 $$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$.…

For a positive integer $n$, let $B_{\ell_1,\dots,\ell_r}(n)$ denote the number of $(\ell_1,\ell_2,\cdots,\ell_r)$-regular multipartitions of $n$. If $\ell_1=\ell_2=\cdots=\ell_r=\ell$, then we denote $B_{\ell_1,\dots,\ell_r}(n)$ as…

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such…

We investigate congruence relations of the form $p_r(\ell m n + t) \equiv 0 \pmod{\ell}$ for all $n$, where $p_r(n)$ is the number of $r$-colored partitions of $n$ and $m,\ell$ are distinct primes.

A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close…

Let $T_\ell(n)$ denote the number of $\ell-$regular partition triples of $n$ and let $p_{\ell, 3}(n)$ enumerates the number of 2--color partition triples of $n$ where one of the colors appear only in parts that are multiples of $\ell$. In…

Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+\beta) \equiv 0 \pmod{\ell}.$…

For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$…

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…

Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of…

Let $ B_{\ell}(n)$ denote the number of $\ell$-regular bipartitions of $n.$ In this article, we prove that $ B_{\ell}(n)$ is always almost divisible by $p_i^j$ if $p_i^{2a_i}\geq \ell,$ where $j$ is a fixed positive integer and…

In this study, we explore the arithmetic properties of $b_{7^k}(n)$ for any $k\geq1$, which enumerates the partitions of $n$ where no part is divisible by $7^k$. By constructing generating functions for $b_{7^k}(n)$ over specific arithmetic…

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$…

In recent work, Amdeberhan and Merca considered the integer partition function $a(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts…

We consider $cp_{a,b,m}(n)$, the number of $(a,b,m)$-copartitions of $n$. We find many infinitelymany congruencesmodulo 2 and 6 for some particular value of $a$, $b$ and $m$.

For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a…

A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite…

Let $a_{r,s}(n)$ denote the number of mutlicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. The paper aims to study arithmetic properties…

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…